Going Beyond the Carothers, Flory and Stockmayer Equation by Including Cyclization Reactions and Mobility Constraints

A challenge in the field of polymer network synthesis by a step-growth mechanism is the quantification of the relative importance of inter- vs. intramolecular reactions. Here we use a matrix-based kinetic Monte Carlo (kMC) framework to demonstrate that the variation of the chain length distribution and its averages (e.g., number average chain length xn), are largely affected by intramolecular reactions, as mostly ignored in theoretical studies. We showcase that a conventional approach based on equations derived by Carothers, Flory and Stockmayer, assuming constant reactivities and ignoring intramolecular reactions, is very approximate, and the use of asymptotic limits is biased. Intramolecular reactions stretch the functional group (FG) conversion range and reduce the average chain lengths. In the likely case of restricted mobilities due to diffusional limitations because of a viscosity increase during polymerization, a complex xn profile with possible plateau formation may arise. The joint consideration of stoichiometric and non-stoichiometric conditions allows the validation of hypotheses for both the intrinsic and apparent reactivities of inter- and intramolecular reactions. The kMC framework is also utilized for reverse engineering purposes, aiming at the identification of advanced (pseudo-)analytical equations, dimensionless numbers and mechanistic insights. We highlight that assuming average molecules by equally distributing A and B FGs is unsuited, and the number of AB intramolecular combinations is affected by the number of monomer units in the molecules, specifically at high FG conversions. In the absence of mobility constraints, dimensionless numbers can be considered to map the time variation of the fraction of intramolecular reactions, but still, a complex solution results, making a kMC approach overall most elegant.

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Chain-Length Distribution Functions during Polymerization

Rubber Chemistry and Technology ◽

10.5254/1.3543518 ◽

1954 ◽

Vol 27 (3) ◽

pp. 622-628 ◽

Cited By ~ 5

Author(s):

W. F. Watson

Keyword(s):

Reaction Mechanism ◽

Chain Length ◽

Length Distribution ◽

Kinetic Scheme ◽

Distribution Functions ◽

Average Chain Length ◽

Chain Length Distribution ◽

Molecular Weights ◽

Measurable Rate ◽

Chain Lengths

Abstract Functions for the distribution of chain lengths of a polymer formed during polymerization have been evaluated in terms of the directly measurable rate and rate of initiation, or the single equivalent measurement of number-average chain length. No details of the reaction mechanism are required, except for the occurrence of termination by combination of polymer radicals. This is in contrast to the usual derivation of distribution functions from the postulated kinetic scheme. The three types of termination are considered, (1) combination absent, (2) combination predominant, and (3) a mixture of combination with other modes of termination. The application to copolymerization is outlined. Relationships between the various average molecular weights are considered.

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Chain-Length Distribution Functions of Polymers after Random Degradation and Cross-Linking, with Particular Reference to Elastomers

Rubber Chemistry and Technology ◽

10.5254/1.3543519 ◽

1954 ◽

Vol 27 (3) ◽

pp. 629-633

Author(s):

W. F. Watson

Keyword(s):

Chain Length ◽

Statistical Treatment ◽

Length Distribution ◽

Distribution Functions ◽

Average Chain Length ◽

Cross Linking ◽

Chain Length Distribution ◽

Polymer Formation ◽

Chain Lengths

Abstract The distribution of chain lengths of polymers on formation, random degradation and random cross-linking, have been derived by a simple statistical treatment. Chain-length distribution functions for all cases are represented by special forms of the expression : Nx/N=(α+β+γ)exp[−(α+β+γ)x] where β is the reciprocal of the average chain length on polymer formation, α is the degree of random degradation, and γ is the degree of cross-linking.

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Network Chain Distribution and Strength of Vulcanizates

Rubber Chemistry and Technology ◽

10.5254/1.3539246 ◽

1969 ◽

Vol 42 (3) ◽

pp. 659-665 ◽

Cited By ~ 13

Author(s):

S. D. Gehman

Keyword(s):

Physical Properties ◽

Chain Length ◽

Random Distribution ◽

Length Distribution ◽

Chain Scission ◽

Point Of View ◽

Average Chain Length ◽

Chain Length Distribution ◽

Network Chain ◽

Chain Molecules

Abstract Physical characteristics of rubber network structures usually enumerated and discussed are network chain density, crosslink functionality, average chain length between crosslinks, entanglements which act somewhat like crosslinks, and free chain ends which are network defects. Chemical factors include structure of the chain molecules, type of crosslinks, whether monosulfide, disulfide or polysulfide, or direct carbon-to-carbon bonds. Side effects of vulcanization reactions such as chain scission or combination of minor quantities of chemical fragments from the vulcanizing system are also recognized. One might think that these variables would be adequate to account for physical properties of elastomers but explanations of strength aspects of vulcanizates are still unsatisfactory. Something is missing in these considerations, that is, the distribution of crosslinks along a main chain or the length sequences of monomer units in network chains. Usually a random distribution is implicitly assumed. If the distribution is always random and nothing can be done about it and it cannot be measured anyway, there may seem to be little point in writing about it. However, an ideally random distribution for all crosslinking systems and polymers seems very improbable. The importance of network chain length distribution for physical properties has been, of course, well recognized in theory. Bueche's calculations showed that viscoelastic resistance to deformation increased markedly with increased crosslink functionality, that is, as more chains are involved in the displacement of a crosslink. His molecular theory of tensile strength was based on the concept of short, overloaded network chains which snapped and transferred their loads to neighboring chains. An alternate point of view is that short chains are detrimental because they do not stress orient as well as longer chains.

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Recent Advances in Controlled Grafting of Elastomers

Rubber Chemistry and Technology ◽

10.5254/1.3536020 ◽

1984 ◽

Vol 57 (3) ◽

pp. 557-582 ◽

Cited By ~ 10

Author(s):

Roderic P. Quirk

Keyword(s):

Chain Length ◽

Graft Copolymers ◽

Length Distribution ◽

Careful Examination ◽

Average Chain Length ◽

Chain Length Distribution ◽

Structure Morphology ◽

Graft Polymers ◽

Exciting Time ◽

Grafting From

Abstract This review describes recent results for preparing graft copolymers with controlled structures. In general, the macromonomer approach appears to be the most promising method for preparation of graft polymers with well defined structures. Not only can macromonomers be prepared using radical, cationic, or anionic polymerization procedures, but the resultant macromonomers can be polymerized using these same methods. Obviously, the macromonomer functionality must be high and well defined. In addition, the average chain length and chain length distribution should be well characterized. In principle, the macromonomer approach can provide comb-type graft copolymers with a random distribution of well characterized graft branches. However, the actual copolymerization of the macromonomers with a variety of comonomers requires careful examination under a variety of reaction conditions before that expectation can be realized. These macromonomers provide an excellent opportunity to examine the effects of chain length on the problem of heterogeneity (i.e., phase-separation) which is expected and found in many grafting systems. “Grafting-from” reactions also should provide graft polymers whose structures are more amenable to prediction and analysis. Like the macromonomer method, it should be possible to eliminate the presence of unwanted homopolymer using this method. However, much more research is required before the generality and value of this method can be evaluated. In conclusion, this era appears to be an exciting time for graft polymerization research. The potential exists for the preparation of model graft polymers with precise structural definition using the methods described herein. This development will, at last, provide graft polymers which can be used to define the relationships between the structure, morphology, and properties of these materials.

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Direct Observation of Chain Lengths and Conformations in Oligofluorene Distributions from Controlled Polymerization by Double Electron-Electron Resonance

10.26434/chemrxiv.10002785 ◽

2019 ◽

Author(s):

Dennis Bücker ◽

Annika Sickinger ◽

Julian D. Ruiz Perez ◽

Manuel Oestringer ◽

Stefan Mecking ◽

...

Keyword(s):

Chain Length ◽

Length Distribution ◽

Catalyst System ◽

Chain Conformation ◽

Controlled Polymerization ◽

Chain Length Distribution ◽

Electron Resonance ◽

Double Electron ◽

The Individual ◽

Double Electron Electron Resonance

Synthetic polymers are mixtures of different length chains, and their chain length and chain conformation is often experimentally characterized by ensemble averages. We demonstrate that Double-Electron-Electron-Resonance (DEER) spectroscopy can reveal the chain length distribution, and chain conformation and flexibility of the individual n-mers in oligo-(9,9-dioctylfluorene) from controlled Suzuki-Miyaura Coupling Polymerization (cSMCP). The required spin-labeled chain ends were introduced efficiently via a TEMPO-substituted initiator and chain terminating agent, respectively, with an in situ catalyst system. Individual precise chain length oligomers as reference materials were obtained by a stepwise approach. Chain length distribution, chain conformation and flexibility can also be accessed within poly(fluorene) nanoparticles.

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GLC determination of chain length distribution in fatty alcohol polyethoxylates and sulfated derivatives

Journal of the American Oil Chemists Society ◽

10.1007/bf02682573 ◽

1966 ◽

Vol 43 (12) ◽

pp. 690-690 ◽

Cited By ~ 5

Author(s):

S Lee ◽

N. A. Puttnam

Keyword(s):

Chain Length ◽

Fatty Alcohol ◽

Length Distribution ◽

Chain Length Distribution

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Chain-length dependent termination in pulsed-laser polymerization, 7. The evaluation of the power-law exponentb from the chain-length distribution in the low frequency (single-pulse) limit for the reference systems styrene and methyl methacrylate in bulk at 25°C

Macromolecular Chemistry and Physics ◽

10.1002/(sici)1521-3935(19990901)200:9<2031::aid-macp2031>3.0.co;2-1 ◽

1999 ◽

Vol 200 (9) ◽

pp. 2031-2039 ◽

Cited By ~ 21

Author(s):

Oskar Friedrich Olaj ◽

Philipp Vana ◽

Andreas Kornherr ◽

Gerhard Zifferer

Keyword(s):

Methyl Methacrylate ◽

Chain Length ◽

Power Law ◽

Length Distribution ◽

Low Frequency ◽

Pulsed Laser ◽

Single Pulse ◽

Reference Systems ◽

Chain Length Distribution ◽

Pulsed Laser Polymerization

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Miniaturized Analysis of Amylopectin Chain Length Distribution by Fluorescence-Assisted Capillary Electrophoresis (FACE) down to Single Starch Granules

10.26434/chemrxiv.13568228 ◽

2021 ◽

Author(s):

amandine pruvost ◽

stanislas helle ◽

nicolas szydlowski ◽

Christian ROLANDO

Keyword(s):

Capillary Electrophoresis ◽

Chain Length ◽

Solid Phase ◽

Potato Starch ◽

Relative Proportion ◽

Length Distribution ◽

Fluorescence Analysis ◽

Starch Granules ◽

Chain Length Distribution ◽

Single Granule

In the present work, we developed a miniaturized method for determining amylopectin chain length distribution (CLD) by fluorescence-assisted capillary electrophoresis (FACE). The method relies on single granule entrapping into capillaries followed by direct starch gelatinization and amylopectin debranching on carbograph-based solid phase extraction (SPE) cartridges. Sample desalting on HypersepTM tips following APTS-labelling and the use of nanovials allowed for the fluorescence analysis of weakly diluted samples. Consequently, method sensitivity was improved by 500-fold which is compatible with the analysis of single potato starch granules. The method was implemented to determine CLD profiles of single starch granules ranging from 50 to 100 µm in diameter. In these experiments, the relative proportion of starch glucans of up to 30 degrees of polymerization (DP) could be quantified.

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